Flux-normalized elastic characteristic amplitudes (${\varphi }_{\pm }=(\sqrt{Z}v\mp \sigma /\sqrt{Z})/2$)

These characteristic wave amplitudes split a one-dimensional elastic wave into forward and backward components. Their instantaneous net elastic-wave energy flux is ${\varphi }_{+}^2-{\varphi }_{-}^2$. For real-part harmonic phasors the mean flux is $(|{\varphi }_{+}{|}^2-|{\varphi }_{-}{|}^2)/2$. In a static graded medium with real positive seismic impedance, put $g=(\log Z{)}^{\prime }/2$; then ${\partial }_x{\varphi }_{\pm }\pm {\beta }^{-1}{\partial }_t{\varphi }_{\pm }=g{\varphi }_{\mp }$. Seismic impedance gradients couple the two directions while preserving net flux.

At zero frequency the graded-layer transfer equation integrates to $\exp [L\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)]$, where $L=\frac{1}{2}\log (Z_b/Z_a)$. Eliminating the outgoing wave amplitude at the left endpoint gives the displayed incoming-to-outgoing scattering matrix. It is real orthogonal, so it preserves total incoming and outgoing flux. It coincides with continuity of velocity and traction at a sharp seismic impedance interface; the profile between the endpoints enters at nonzero frequency.